G² continuity of NURBS curves

This page defines geometric continuity between curve segments and derives the endpoint curvature formulas needed to join Bézier, B-spline, and NURBS curves smoothly, then applies them to airfoil parametrization at the leading and trailing edges, closing with a comparison to Mykhaskiv et al. [2018]. The basis-function and endpoint-derivative definitions are covered on the Bézier, B-spline, and NURBS pages; each curvature section below recalls only what it needs.

Motivation

In many science and engineering applications it is necessary to develop a smooth geometric parametrization. The level of smoothness of a curve or surface can be measured by its degree of geometric continuity:

• \(G^0\), or position continuity,

• \(G^1\), or tangency continuity,

• \(G^2\), or curvature continuity,

• and so on.

When the geometry is parametrized by a single Bézier, B-spline, or NURBS curve, it is straightforward to ensure the desired level of continuity at the interior points: the curve is polynomial or rational there, hence smooth, subject to the knot-multiplicity guarantees already stated on the family pages. However, when the geometry is described with more than one curve segment, it is necessary to pay special attention to the continuity at the connection points — for instance, where the pressure and suction sides of an airfoil meet at the leading and trailing edges.

For a regular space curve \(\mathbf C(u)\), parametrized by \(u\), define the unit tangent and curvature vector

(1)\[\mathbf T = \frac{\mathbf C'}{\|\mathbf C'\|}, \qquad \mathbf K = \frac{\mathbf C'' - \mathbf T(\mathbf T\cdot\mathbf C'')}{\|\mathbf C'\|^2}.\]

The curvature magnitude and radius of curvature are

(2)\[\kappa = \|\mathbf K\| = \frac{\|\mathbf C''\times\mathbf C'\|}{\|\mathbf C'\|^3}, \qquad \rho = \frac{1}{\kappa}.\]

Attaining \(G^2\) continuity at a join means matching both the oriented unit tangent and the curvature vector there, not merely the scalar curvature or radius of curvature. Suppose \(\mathbf C_A(1)\) joins \(\mathbf C_B(0)\), with both segments following the intended direction of traversal:

Continuity	Requirement at the join
\(G^0\)	\(\mathbf C_A(1)=\mathbf C_B(0)\)
\(G^1\)	\(G^0\) and \(\mathbf T_A(1)=\mathbf T_B(0)\)
\(G^2\)	\(G^1\) and \(\mathbf K_A(1)=\mathbf K_B(0)\)

Equal curvature magnitudes \(\|\mathbf K_A\|=\|\mathbf K_B\|\) are not enough: the two curvature vectors can point in opposite directions. For example, an endpoint velocity \((1,0)\) paired with accelerations \((0,2)\) and \((0,-2)\) gives the same tangent and the same unsigned curvature on both sides, but opposite curvature vectors. Matching a common radius of curvature \(\rho=1/\kappa\) is therefore sufficient for \(G^2\) only once matching position and oriented tangent, and matching curvature direction, are also imposed.

For regular segments with one-sided second derivatives, an equivalent statement of \(G^2\) is that there exist \(\lambda>0\) and \(\mu\in\mathbb R\) with

(3)\[\mathbf C'_B(0) = \lambda\, \mathbf C'_A(1), \qquad \mathbf C''_B(0) = \lambda^2\, \mathbf C''_A(1) + \mu\, \mathbf C'_A(1),\]

in addition to coincident endpoints. These follow from the chain rule for an orientation-preserving reparametrization \(u\mapsto\tilde u(u)\) with \(\tilde u'(0)=\lambda\) and \(\tilde u''(0)=\mu\): unlike parametric \(C^2\) continuity, which forces \(\lambda=1\) and \(\mu=0\), geometric continuity allows the two segments to run at different parameter speeds. This is the sense in which the phrase “continuity of the radius of curvature” is used below and in the original note: it always means matching the vector quantities above, not merely their norms.

The usual approach to ensure smoothness at the point connecting two curves is to add extra control points at suitable locations. This is done, for instance, to ensure \(G^2\) continuity at the points connecting the pressure and suction surfaces of an airfoil. The objective of the rest of this page is to derive expressions for the endpoint curvature of Bézier and B-spline curves, use them to place the additional control points required for \(G^2\) continuity between two spline segments, apply the result to an airfoil parametrization that meets \(G^2\) continuity at the leading and trailing edges, and finally compare these relations with the expression used for the 2D blade parametrization in Mykhaskiv et al. [2018], suggesting a correction for the case of B-spline curves.

Bézier curve endpoint curvature formulas

Recall the Bézier curve (1) and its Bernstein basis from the Bézier page. Its endpoint value, first derivative, and second derivative are

\[\mathbf C(u=0)=\mathbf P_0, \qquad \mathbf C(u=1)=\mathbf P_n,\]

\[\dot{\mathbf C}(u=0) = n(\mathbf P_1-\mathbf P_0), \qquad \dot{\mathbf C}(u=1) = n(\mathbf P_n-\mathbf P_{n-1}),\]

\[\ddot{\mathbf C}(u=0) = n(n-1)\big[(\mathbf P_2-\mathbf P_0)-2(\mathbf P_1-\mathbf P_0)\big],\]

\[\ddot{\mathbf C}(u=1) = n(n-1)\big[(\mathbf P_{n-2}-\mathbf P_n)-2(\mathbf P_{n-1}-\mathbf P_n)\big].\]

See Piegl and Tiller [1997], pp. 9–25, for proofs of these endpoint formulas. Inserting the first and second derivatives into the radius of curvature definition (2), and noting that the cross product of two parallel vectors is zero, gives the endpoint curvature formulas already stated as (7):

(4)\[\kappa(u=0) = \left(\frac{n-1}{n}\right) \frac{\big\|(\mathbf P_2-\mathbf P_0)\times(\mathbf P_1-\mathbf P_0)\big\|} {\|\mathbf P_1-\mathbf P_0\|^3},\]

(5)\[\kappa(u=1) = \left(\frac{n-1}{n}\right) \frac{\big\|(\mathbf P_{n-2}-\mathbf P_n)\times(\mathbf P_{n-1}-\mathbf P_n)\big\|} {\|\mathbf P_{n-1}-\mathbf P_n\|^3}.\]

B-spline curve endpoint curvature formulas

Recall the B-spline curve (1) and its Cox–de Boor basis from the B-spline page. Its endpoint value, first derivative, and second derivative are

\[\mathbf C(u=0)=\mathbf P_0, \qquad \mathbf C(u=1)=\mathbf P_n,\]

\[\dot{\mathbf C}(u=0) = \frac{p}{u_{p+1}}(\mathbf P_1-\mathbf P_0), \qquad \dot{\mathbf C}(u=1) = \frac{p}{1-u_n}(\mathbf P_n-\mathbf P_{n-1}),\]

\[\ddot{\mathbf C}(u=0) = \frac{p(p-1)}{u_{p+1}}\left[ \frac{1}{u_{p+2}}(\mathbf P_2-\mathbf P_0) -\left(\frac{1}{u_{p+1}}+\frac{1}{u_{p+2}}\right)(\mathbf P_1-\mathbf P_0) \right],\]

\[\ddot{\mathbf C}(u=1) = \frac{p(p-1)}{1-u_n}\left[ \frac{1}{1-u_{n-1}}(\mathbf P_{n-2}-\mathbf P_n) -\left(\frac{1}{1-u_n}+\frac{1}{1-u_{n-1}}\right)(\mathbf P_{n-1}-\mathbf P_n) \right].\]

See Piegl and Tiller [1997], pp. 81–100, for a proof of these endpoint formulas. As before, inserting the first and second derivatives into (2) and cancelling the cross product of the parallel tangential term gives the endpoint curvature formulas already stated as (7):

(6)\[\kappa(u=0) = \left(\frac{p-1}{p}\right)\left(\frac{u_{p+1}}{u_{p+2}}\right) \frac{\big\|(\mathbf P_2-\mathbf P_0)\times(\mathbf P_1-\mathbf P_0)\big\|} {\|\mathbf P_1-\mathbf P_0\|^3},\]

(7)\[\kappa(u=1) = \left(\frac{p-1}{p}\right)\left(\frac{1-u_n}{1-u_{n-1}}\right) \frac{\big\|(\mathbf P_{n-2}-\mathbf P_n)\times(\mathbf P_{n-1}-\mathbf P_n)\big\|} {\|\mathbf P_{n-1}-\mathbf P_n\|^3}.\]

NURBS curve endpoint curvature formulas

Recall the NURBS curve (1) and its rational basis from the NURBS page. Its endpoint value, first derivative, and second derivative are

\[\mathbf C(u=0)=\mathbf P_0, \qquad \mathbf C(u=1)=\mathbf P_n,\]

\[\dot{\mathbf C}(u=0) = \left(\frac{p}{u_{p+1}}\right)\left(\frac{w_1}{w_0}\right)(\mathbf P_1-\mathbf P_0), \qquad \dot{\mathbf C}(u=1) = \left(\frac{p}{1-u_n}\right)\left(\frac{w_{n-1}}{w_n}\right)(\mathbf P_n-\mathbf P_{n-1}),\]

\[\ddot{\mathbf C}(u=0) = \frac{p(p-1)}{u_{p+1}}\left[ \frac{1}{u_{p+2}}\frac{w_2}{w_0}(\mathbf P_2-\mathbf P_0) -\left(\frac{1}{u_{p+1}}+\frac{1}{u_{p+2}}\right)\frac{w_1}{w_0}(\mathbf P_1-\mathbf P_0) \right]\]

\[{}+\frac{2p^2}{u_{p+1}^2}\frac{w_1}{w_0}\left(1-\frac{w_1}{w_0}\right)(\mathbf P_1-\mathbf P_0),\]

\[\ddot{\mathbf C}(u=1) = \frac{p(p-1)}{1-u_n}\left[ \frac{1}{1-u_{n-1}}\frac{w_{n-2}}{w_n}(\mathbf P_{n-2}-\mathbf P_n) -\left(\frac{1}{1-u_n}+\frac{1}{1-u_{n-1}}\right)\frac{w_{n-1}}{w_n}(\mathbf P_{n-1}-\mathbf P_n) \right]\]

\[{}+\frac{2p^2}{(1-u_n)^2}\frac{w_{n-1}}{w_n}\left(1-\frac{w_{n-1}}{w_n}\right)(\mathbf P_{n-1}-\mathbf P_n).\]

See Piegl and Tiller [1997], pp. 117–127, for a proof of these endpoint formulas. Inserting the first and second derivatives into (2), and again cancelling the cross product of the parallel tangential term, gives the endpoint curvature formulas already stated as (6):

(8)\[\kappa(u=0) = \left(\frac{p-1}{p}\right)\left(\frac{u_{p+1}}{u_{p+2}}\right) \left(\frac{w_0w_2}{w_1^2}\right) \frac{\big\|(\mathbf P_2-\mathbf P_0)\times(\mathbf P_1-\mathbf P_0)\big\|} {\|\mathbf P_1-\mathbf P_0\|^3},\]

(9)\[\kappa(u=1) = \left(\frac{p-1}{p}\right)\left(\frac{1-u_n}{1-u_{n-1}}\right) \left(\frac{w_nw_{n-2}}{w_{n-1}^2}\right) \frac{\big\|(\mathbf P_{n-2}-\mathbf P_n)\times(\mathbf P_{n-1}-\mathbf P_n)\big\|} {\|\mathbf P_{n-1}-\mathbf P_n\|^3}.\]

This equation applies to NURBS curves and includes Bézier and B-spline curves as the special cases above.

Application to airfoil parametrization

The endpoint curvature formulas for NURBS curves can be used to specify the radius of curvature at the leading and trailing edges of an airfoil and ensure \(G^2\) continuity between the upper and lower surfaces.

Leading edge

Consider the construction of the suction side at the leading edge. The suction side is defined by the set of control points \(\{\mathbf P_0,\mathbf P_2,\ldots,\mathbf P_{n-2},\mathbf P_n\}\), where \(\mathbf P_0\) is the start point of the camber line and \(\mathbf P_2\) is computed from the thickness distribution. To impose a radius of curvature \(\rho=1/\kappa\) at the leading edge, insert an additional control point \(\mathbf P_1\) at distance \(\ell_0=\|\mathbf P_1-\mathbf P_0\|\) from \(\mathbf P_0\) along a chosen unit direction \(\mathbf t\),

\[\mathbf P_1 = \mathbf P_0 + \ell_0\,\mathbf t, \qquad \ell_0>0.\]

In the original airfoil construction, \(\mathbf t\) is the curve’s unit tangent direction at \(u=0\), which for this construction happens to be normal to the camber line. The distance \(\ell_0\) is found by solving (8), with \(\mathbf P_1-\mathbf P_0=\ell_0\mathbf t\) and \(\mathbf t\times\mathbf t=\mathbf 0\):

(10)\[\ell_0^2 = \rho\left(\frac{p-1}{p}\right)\left(\frac{u_{p+1}}{u_{p+2}}\right) \left(\frac{w_0w_2}{w_1^2}\right)\, \big\|(\mathbf P_2-\mathbf P_0)\times\mathbf t\big\|.\]

This equation applies to NURBS curves and includes Bézier and B-spline curves as special cases. If the component of \(\mathbf P_2-\mathbf P_0\) perpendicular to \(\mathbf t\) vanishes, no finite positive \(\ell_0\) produces this curvature. Repeating the process for the pressure side using the same radius of curvature, oriented tangent, and curvature direction ensures \(G^2\) continuity at the leading edge.

Trailing edge

The procedure for the trailing edge is analogous. Here, \(\mathbf P_n\) is the endpoint of the camber line and \(\mathbf P_{n-2}\) is computed from the thickness distribution. Choose a unit direction \(\mathbf t_{\rm back}\) pointing from \(\mathbf P_n\) toward the preceding control point, and insert an additional control point

\[\mathbf P_{n-1} = \mathbf P_n + \ell_1\,\mathbf t_{\rm back}, \qquad \ell_1>0.\]

The curve’s forward tangent at \(u=1\) then points along \(-\mathbf t_{\rm back}\). The distance \(\ell_1\) is found by solving (9):

(11)\[\ell_1^2 = \rho\left(\frac{p-1}{p}\right)\left(\frac{1-u_n}{1-u_{n-1}}\right) \left(\frac{w_nw_{n-2}}{w_{n-1}^2}\right)\, \big\|(\mathbf P_{n-2}-\mathbf P_n)\times\mathbf t_{\rm back}\big\|.\]

This equation applies to NURBS curves and includes Bézier and B-spline curves as special cases. Repeating the process for the pressure side using the same radius of curvature ensures \(G^2\) continuity at the trailing edge.

Both constructions only fix a control polygon that reproduces a chosen radius of curvature; verifying full \(G^2\) continuity between the two sides still requires checking that positions coincide and that the oriented tangents and curvature vectors agree, as in the Motivation section above, including a consistent traversal direction for the closed profile.